Self-Affine Tilings with Several Tiles, I

Applied and Computational Harmonic Analysis 7, 211–238 (1999) Article ID acha.1998.0268, available online at http://www.idealibrary.com on

Self-Affine Tilings with Several Tiles, I Karlheinz Gröchenig and Andrew Haas Department of Mathematics, The University of Connecticut, Storrs, Connecticut, 06269-3009 E-mail: [email protected]; [email protected]

and Albert Raugi Départment de Mathématiques, Université de Rennes I, Campus de Beaulieu, F-35000 Rennes Cedex, France E-mail: [email protected]

Communicated by W.R. Madych Received December 24, 1997; revised July 22, 1998

The tilings of Rd by a finite number of lattice translates of self-affine prototiles are studied in their own right and as they relate to multiwavelet bases of L2 (Rd ).  1999 Academic Press

1. INTRODUCTION

The mathematical literature provides several notions of “self-similar tilings,” which differ mainly by the group of motions that act on the prototiles [11, 15, 21, 22, 24]. The vague label is used to describe such strikingly different objects as the aperiodic Penrose tilings [21] and the periodic tilings obtained from the “twin-dragon fractal” [10, 19, 25]. In this paper we study self-affine tilings of Rd by a finite set of compact prototiles Ti , which tile Rd by translations in a lattice 3 ⊆ Zd . More precisely, a finite collection of sets d T = {Ti ⊆ Rd }M i=1 , consisting of sets that are either compact or empty, is said to 3-tile R if

setting T =

SM

i=1 Ti

Ti ∩ Tj ∼ =∅ [

for i 6= j,

(k + T ) ∼ = Rd

(1)

(2)

k∈3

and T ∩ (k + T ) ∼ =∅

for k ∈ 3, k 6= 0.

(3)

For two sets X, Y ⊆ Rd we write X ∼ = Y if their symmetric difference X 4 Y = (X \ Y ) ∪ (Y \ X) has Lebesgue measure 0, i.e., |X 4 Y | = 0. As usual, the sum of sets X, Y ⊆ Rd 211 1063-5203/99 $30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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is defined by X + Y = {z = x + y | x ∈ X, y ∈ Y } if both X and Y are nonempty and X + Y = ∅ otherwise. Allowing prototiles to be empty makes it easier to access an important facet of the tiling problem, which is easily overlooked when all tiles are assumed to be nonempty. A compact set T satisfying (2) and (3) is called a 3-tile. It follows from (3) that if T is a Zd tile, then T has measure one. Sets Ti ⊆ Rd , 1 ≤ i ≤ M, that are either compact or empty form a (A, 0)-self-affine collection if there is an integer-valued d × d-matrix A with all eigenvalues of modulus greater than one and finite (possibly empty) sets 0ij ⊆ Zd , i, j = 1, . . . , M, so that ATi =

M [

(0ij + Tj )

for i = 1, . . . , M,

(4)

j =1

and for any i, j, k (β + Ti ) ∩ (γ + Tj ) ∼ =∅

for β ∈ 0ki , γ ∈ 0kj and i 6= j or β 6= γ .

(5)

The matrix A is usually called a dilation or an expanding matrix and the set 0 = {0ij } is a digit set. Note that the essential disjointness in (5) needs only hold for different sets in the same equation of (4). If sets T = {Ti , i = 1, . . . , M} form a self-affine collection for some A and 0 and 3-tile Rd , then we call them a self-affine 3-tiling set with M prototiles, abbreviated SAT. The stipulation that the prototiles are positioned in the tiling by translation in a lattice is rather restrictive and excludes many self-similar (self-affine) tilings that appear in the literature [11, 15, 21, 22, 24]. The SATs with one prototile have been well studied. Interest in them became more intense after the discovery of a connection to wavelet theory [10]. It is known through papers of Gröchenig and Haas, Lagarias and Wang, and Conze et al. [2, 3, 9, 18] in dimensions d = 1, d = 2, d ≥ 3, respectively, that if the single set 0 = 011 is a complete set of coset representatives for the group Zd /AZd , then there is a compact self-affine set Q solving (4) and a lattice 3 ⊆ Zd , for which Q is a 3-tile. In dimension d = 1 this lattice is explicitly 3 = nZ, where n is the greatest common divisor of elements in 0 (where we assumed without loss of generality that 0 ∈ 0). In dimension d ≥ 2 it is known how to determine 3 in principle [3, 18], but the problem of characterizing 3 explicitly in terms of A and 0 is still unresolved. Also, given a dilation matrix A it is not yet known if there exists a digit set 0 for which 3 = Zd [17]. In short, even in the simplest case M = 1 of a single prototile there are still large gaps in the theory. The focus of this paper is the class of SATs with M > 1 prototiles. Given A and 0 = {0ij , i, j = 1, . . . , M}, we show there exist finitely many (A, 0)-self-affine collections and we give necessary conditions for a collection to be a 3-tiling, with special attention to the case 3 = Zd . It is then possible to establish a connection to wavelet theory, similar to the one in [10]; that is, between SATs with several prototiles and multiwavelet bases. We show that in general an SAT determines a multiwavelet basis of L2 (Rd ) and vice versa. This provides the first systematic construction of multiwavelet bases in higher dimensions with arbitrary dilation matrices. Solutions of the dilation equations (4) can be described by means of digit expansions, in which the allowable sequence of digits in an expansion resembles the orbit of a point under

SELF-AFFINE TILINGS WITH SEVERAL TILES I

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a subshift of finite type. As in the one tile case, M = 1, (4) has exactly one solution with all compact prototiles. In contrast to the one tile case, when some prototiles are allowed to be empty there may exist several solutions. The familiar approach of constructing a selfsimilar set as the fixed point of an associated iterated function system will only produce the solution of equations (4) with all prototiles compact, but a slight modification, for which it is possible to have multiple attracting fixed points and also attracting periodic orbits, will suffice to generate all solutions. We define the notion of a standard digit set 0, resembling the standard digit sets introduced in [10, 16]. Elementary arguments show that for a self-affine collection to be a Zd -tiling, 0 must be a standard digit set. As is shown by many examples, this condition is far from sufficient. The theory of Markoff chains over a finite state space is used to analyze the self-affine collection associated to a standard digit set and results in a necessary condition for Zd -tiling in terms of a Markoff chain determined by the overall structure of the dilation equations. Other conditions are given that take account of both the overall structure of the equations and the specific digits involved. If the characteristic functions χTi (x) are combined in a column vector 8(x), then the dilation equation (4) can be written as 8(x) =

X

Ck 8(Ax − k),

k∈Zd

where Ck is an M × M matrix with entries being either 0 or 1. Such equations are among the main objects in wavelet theory and are called vector-valued scaling relations or vector-valued refinement equations. If a self-affine collection is a Zd -tiling, we show that 8 gives rise to a multiresolution analysis with multiplicity M. Conversely, to any multiresolution analysis whose basis functions are characteristic functions corresponds a self-affine Zd -tiling. To every self-affine Zd -tiling we then construct a particular orthonormal basis of L2 (Rd ), a so-called wavelet basis. These results complement the theory of multiwavelets [7, 8, 13] with concrete examples and extend the work in [10] to higher multiplicities. The paper is organized as follows: The first section is an example “zoo” and by strolling through it the reader should get a better sense of the objects under consideration. It was the confusing and mysterious variety of examples that initially sparked our interest, and we hope that they will stimulate the reader’s curiosity. In Section 3 we construct and classify the general solutions of (4) and in Section 4 we derive a necessary condition for a solution of (4) to be a Zd -tiling. Section 5 establishes the relation to the theory of multiwavelets and constructs a class of orthonormal bases for L2 (Rd ) starting from a SAT. In a sequel to this paper we will use Fourier analytic methods and the theory of the transfer operator to study SATs in a more systematic fashion. While preparing the final version of this manuscript we became aware of an interesting preprint [6] of Flaherty and Wang titled “Haar-type multiwavelet bases and self-affine multi-tiles” which overlaps slightly with our Section 5 in results, but not in its approach. 2. EXAMPLES

In order to demonstrate the almost confusing wealth of different phenomena, the examples will be presented first, with the more involved details left to the end of the section.

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In Section 3 it will be shown that the dilation equation (4) always has a unique (maximal) solution for which each of the Ti ’s is compact. This solution is denoted Q = {Qi }. The dilation equations have the trivial solution Ti = ∅, i = 1, . . . , M, and often other solutions as well. We assume, throughout this discussion, that for some i, Ti 6= ∅. 2.1. A General Example We begin with a construction that produces nontrivial examples in any dimension d. It is based on the existence of lattice tilings with a single tile. Let 0 = {γ1 , . . . , γq } be a set of coset representatives for the group Zd /AZd where A is expanding and | det A| = q. By the theorem [3, 18] there is a lattice 3 ⊆ Zd and a unique compact solution Q to the equation AT = 0 + T , so that 3 + Q is a tiling of Rd . Now choose arbitrary αi ∈ 3, i = 1, . . . , q, set γij = γi + Aαi − αj and 0ij = {γij }. The dilation equations (4) then have the solution Q = {Qi } with Qi = A−1 Q + A−1 γi + αi . Sq Since the sets β + Q = i=1 (β + A−1 γi + A−1 Q) are all disjoint for β ∈ 3, so are the sets Qi . Thus insofar as we understand Q, we can construct a self-affine 3-tiling set Q = {Qi }. In particular the choice 0 = {γ1 = (0, 0), γ2 = (0, 1)}, α1 = (0, 0), and α2 = (1, 1) for the dilation matrix   1 −1 A= 1 1 yields the equations AT1 = T1 ∪ ((−1, −1) + T2 ), AT2 = ((0, 3) + T1 ) ∪ ((−1, 2) + T2 ).

(6)

They have a solution which consists of two contracted, translated copies of the well-known twin dragon, cf. Fig. 1. Since the solution of AT = 0 + T yields a Z2 -tiling [10], by the above argument Q also yields a Z2 -tiling of the plane.

FIGURE 1

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215

2.2. Examples with d = 1 and M > 1 For simplicity we mainly consider examples with M = 2 or M = 3 and Ax = 2x or Ax = 3x in dimension d = 1. Although this may seem a very special case, it already shows the puzzling variety of phenomena in the tiling problem with several tiles. E XAMPLE 1. 2T1 = a + T2 , 2T2 = T1 ∪ (b + T1 ).

(7)

Taking measure we have 2|T1 | = |T2 | and 2|T2 | ≤ 2|T1 |. It follows that any solution has measure zero, and certainly these sets do not tile R. E XAMPLE 2. 3T1 = T1 ∪ (1 + T1 ) ∪ (2 + T1 ) ∪ T3 , 3T2 = (2 + T2 ) ∪ (3 + T2 ) ∪ (4 + T2 ) ∪ T4 , 3T3 = (−2 + T3 ) ∪ (−4 + T3 ) ∪ (1 + T4 ),

(8)

3T4 = 2 + T4 . These equations have a number of different sets of solutions in each of which T3 and T4 are sets of measure zero. The simplest solutions are: T1 = [0, 1], Ti = ∅ for i 6= 1; T2 = [1, 2], Ti = ∅ for i 6= 2; and T1 = [0, 1], T2 = [1, 2], Ti = ∅, i = 3, 4. The first two solutions are Z-tilings, whereas the third one is a 2Z-tiling set. There are three other solutions for which the Ti are either compact or empty. For the “maximal” solution with all Ti compact Q4 = {1}, Q3 is a Cantor set, and Q1 and Q2 are fractal sets containing the intervals [0, 1] and [1, 2], respectively. As in (7), the form of the dilation equations, without consideration of the particular digits, forces some prototiles to have measure zero. More elaborate instances of this behavior will be considered in Section 4. E XAMPLE 3. For q ∈ Z consider the equations qTi =

q [

(αij + Ti )

for i = 1, . . . , M, αij ∈ Z.

(9)

j =1

The equations decouple and an area argument shows that each equation, and hence the whole set of them, determines a self-affine collection. Observe that some of the prototiles may be chosen to be empty. However, if two Ti ’s have positive measure, then T = {Ti } is not a Z-tiling set. The question of when a set is a 3-tiling set for a lattice 3 is subtle. In the next example we look at a special case where no such 3 exists. E XAMPLE 4. Assume that a, c ≡ 1(mod 2) and b ≡ 0(mod 2) and consider 2T1 = T2 ∪ (a + T2 ), 2T2 = (b + T1 ) ∪ (c + T1 ).

(10)

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In this example there is a unique compact solution which is a self-affine collection. In fact, multiplying the equations by 2 and substituting (10) for 2Ti , we obtain the decoupled equations 4T1 = (b + T1 ) ∪ (c + T1 ) ∪ (b + 2a + T1 ) ∪ (c + 2a + T1 ), 4T2 = (2b + T2 ) ∪ (2c + T2 ) ∪ (a + 2b + T2 ) ∪ (a + 2c + T2 ).

(11)

From our choice of digits a, b, c it is easily deduced that the digit sets 010 = {b, c, 2a + b, 2a + c} and 020 = {2b, 2c, a + 2b, a + 2c} are both congruent to Z/4Z. Therefore by the tiling theorem [9, Theorem 2.3], both Q1 and Q2 are lattice tiles. Consequently, Q = Q1 ∪ Q2 has measure > 1 and is not a Z-tile. Then {Qi } is not a self-affine Z-tiling set either. The simplest choice of digits is a = 1, b = 0, c = 1, which results in Q1 = Q2 = [0, 1]. When a = 1, b = 1, c = 2, we get Q1 = [1/3, 4/3] and Q2 = [2/3, 5/3]. In general, one obtains tiling sets of infinite connectivity. For the particular choice a = 1, b = 2, c = 5 we show that {Qi } is not a lattice tiling for any lattice 3, although each Qi is a Z-tile by itself. In this case 010 = {2, 4, 5, 7} and 020 = {4, 5, 10, 11}. Since by the digit expansion theorem [10], see also Section 3, x ∈ Qi if and P −j 0 only if x = ∞ j =1 4 j , j ∈ 0i , we deduce that Q1 ⊆ [2/3, 7/3] and Q2 ⊆ [4/3, 11/3]. Again by [9, Theorem 2.3], Z + Qi ∼ = R and thus I = [4/3, 7/3] ⊆ Q1 ∪ (1 + Q1 ) and I ⊆ Q2 ∪ (−1 + Q2 ) ∪ (−2 + Q2 ). It is easy to see that |I ∩ ( + Qi )| > 0 for i = 1, 2 and the corresponding translates . On the other hand, if Q1 ∪ Q2 is a 3-tile, then necessarily 3 = 2Z, since |Q1 ∪ Q2 | ≤ 2. This implies that I ⊆ Q2 ∪ (−2 + Q2 ) and thus |I ∩ (−1 + Q2)| = 0 yields a contradiction. We conjecture that in general, if 0i0 does not consist of consecutive integers, then Q cannot be a lattice tiling set. E XAMPLE 5. 2T1 = T2 ∪ (2 + T2 ), 2T2 = T1 ∪ (1 + T1 ) ∪ (2 + T2 ).

(12)

The unique compact solution is Q1 = Q2 = [0, 2]. The first equation expresses 2Q1 as a disjoint union of translates of Q2 , but in the second equation 2Q2 is a union of overlapping intervals. Thus (4) and (5) are only satisfied for the first equation, and Q is not a self-affine collection. Nonetheless they satisfy (2) and (3) with 3 = 2Z. E XAMPLE 6. 2T1 = T1 ∪ (−1 + T3 ), 2T2 = (2 + T1 ) ∪ (1 + T3 ),

(13)

2T3 = (4 + T1 ) ∪ (4 + T2 ). The unique compact solution is Q1 = [0, 1], Q2 = [1, 2], and Q3 = [2, 3]. Thus Q is a self-affine 3Z-tiling set. This example will come back to haunt us. E XAMPLE 7. 2T1 = T1 ∪ T2 , 2T2 = (1 + T1 ) ∪ (3 + T2 ).

(14)

SELF-AFFINE TILINGS WITH SEVERAL TILES I

217

FIGURE 2

This is the most interesting and complicated of the one-dimensional examples we consider. We show that the compact solution Q is a self-affine Z-tiling set at the end of this section. See Fig. 2. The more general case 2T1 = T1 ∪ (a + T2 ), 2T2 = (b + T1 ) ∪ (c + T2 ) is already beyond the scope of this paper and will be considered in the second part. It is worth noting that when a = 0, b = 1, c = 1, the solution is Q1 = [0, 1/2], Q2 = [1/2, 1], a self-affine Z-tiling set. On the other hand, for a = 1, b = 1, c = 0, yields Q1 = [0, 1], Q2 = [0, 1], which does not satisfy (1), although it is a self-affine collection. While the digit sets are very similar, the second example results in an obvious redundancy. 2.3. Examples with d = 2 and M = 2 E XAMPLE 8. Let

 A=

2 0

0 2



and consider AT1 = T1 ∪ ((1, 0) + T1 ) ∪ ((1, 1) + T1 ) ∪ ((1, 0) + T2 ), AT2 = T2 ∪ ((0, 1) + T2 ) ∪ ((1, 1) + T2 ) ∪ ((0, 1) + T1 ).

(15)

These equations yield triangles as a solution; Q1 having vertices (0, 0), (1, 0), (1, 1), and Q2 having vertices (0, 0), (0, 1), (1, 1). Q is a Z2 -tiling set related to the very basic square tiling of the plane. Another example which is illustrated in Fig. 3 with   1 1 A= −1 1 is AT1 = T1 ∪ T2 , AT2 = ((1, 0) + T2 ) ∪ T3 , AT3 = ((1, 0) + T1 ) ∪ ((1, 0) + T3 ). Since T = T1 ∪ T2 ∪ T3 satisfies AT = T ∪ ((1, 0) + T ), the equation for the “twin dragon,” Q is a Z2 -tiling set. S More generally one can choose the digits so that for each j , M i=1 0ij = 0 is a fixed set congruent to Z2 /AZ2 . Computing gives

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FIGURE 3

A

M [

! Ti

=

i=1

=

M [

M [

i=1

j =1

M [

! 0ij + Tj

=

M [

M [

j =1

i=1

M [

(0 + Tj ) = 0 +

j =1

! 0ij + Tj

! Tj .

j =1

S With T = M i=1 Ti we have AT = 0 + T and the tiling theorem in [3] implies that the solution T is a self-affine tile for Rd . Still we do not know whether (1) or (5) are satisfied. In fact, there are examples, cf. Examples 3 and 7 with a = 1, b = 1, c = 0, for which they are not. It would be interesting to know additional conditions necessary to guarantee that these examples work. E XAMPLE 9. Again

 A=

2 0 0 2



and consider AT1 = T1 ∪ ((1, 0) + T1 ) ∪ ((1, 1) + T1 ) ∪ ((0, −1) + T2 ), AT2 = ((2, 3) + T1 ) ∪ ((1, 1) + T2 ) ∪ ((2, 2) + T2 ) ∪ ((1, 2) + T2 ).

(16)

Q1 and Q2 are triangles with vertices (0, 0), (1, 0), (1, 1) and (1, 1), (1, 2), (2, 2), respectively, and Q is a self-affine Z2 -tiling set. This example can be derived from (15) in much the same way as the examples in (9) are constructed from lattice tilings with one tile, i.e., the prototiles have been translated by elements of the lattice 3. We leave the formalities to the reader. E XAMPLE 10. Let

 A=

and

1 1

−1 1



SELF-AFFINE TILINGS WITH SEVERAL TILES I

219

FIGURE 4

AT1 = T1 ∪ T2 , AT2 = ((0, 1) + T1 ) ∪ ((1, 0) + T2 ).

(17)

The compact solution is a self-affine Z2 -tiling set, as will be discussed in the next section. This tiling resembles the well-known twin dragon but cannot, to our knowledge, be obtained from it by any simple maneuver. See Fig. 4. E XAMPLE 11. Figure 5 shows the maximal solution of AT1 = T1 ∪ ((0, 1) + T1 ) ∪ T2 , AT2 = ((1, 0) + T1 ) ∪ ((1, 0) + T2 ) ∪ ((0, 1) + T2 ) with respect to the dilation matrix  A=

1 1 −2 1

FIGURE 5

 .

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FIGURE 6

Note that one tile Q1 is connected, whereas the other tile Q2 is not. The tile Q = Q1 ∪ Q2 is a solution of AQ = Q ∪ ((1, 0) + Q) ∪ ((0, 1) + Q). Q and its translates by (0, 1) and (1, 0) are depicted in Fig. 6. E XAMPLE 12. Figure 7 shows a bizarre solution of AT1 = T1 ∪ ((3, 0) + T1 ) ∪ ((1, 1) + T1 ) ∪ ((0, −1) + T2 ), AT2 = ((2, 3) + T1 ) ∪ ((1, 1) + T2 ) ∪ ((2, 2) + T2 ) ∪ ((1, 2) + T2 ) with

 A=

2 0

0 2

 .

It seems that Q tiles, but we do not have a formal proof for this. 2.4. Details In this section we prove that the self-affine collections in Examples 7 and 10 are Zd -tiling sets. We shall use a direct elementary method that is based on the geometric interpretation

FIGURE 7

SELF-AFFINE TILINGS WITH SEVERAL TILES I

221

of the transfer operator in [12]. We hope that in view of these calculations the reader will appreciate the more systematic approach by means of Fourier analysis and the transfer operator which is the subject of Part II. S Suppose that T is an (A, 0)-self-affine collection, | det A| = q, and that Zd + M i=1 Ti ≡ Rd . Define the non-negative numbers aij (k) = |Ti ∩ (k + Tj )|

for i = 1, . . . , M, and k ∈ Zd .

(18)

To prove that the self-affine collection T is a Zd -tiling, it is sufficient to show that aij (k) = 0

if i 6= j or k 6= 0.

(19)

The following two identities which follow from the self-similarity (4) will prove useful, aij (k) = aj i (−k)

(20)

and qaij (k) =

M X X M X X

alm (β − α + Ak),

(21)

l=1 m=1 α∈0il β∈0jm

where a sum over an empty set equals 0 by definition. Equation (20) is obvious, (21) follows by computation from q|Ti ∩ (k + Tj )| = |ATi ∩ (Ak + ATj )| M ! ! M [ [ 0il + Tl ∩ Ak + 0j m + Tm = l=1

=

M M X X

m=1

X X

|Tl ∩ (β − α + Ak + Tm )|

l=1 m=1 α∈0il β∈0jm

=

M X X M X X

alm (β − α + Ak).

l=1 m=1 α∈0il β∈0jm

In Example 7, 2T1 = T1 ∪ T2 , 2T2 = (1 + T1 ) ∪ (3 + T2 ); (21) takes the form of the following four equations: 2a11(k) = a11 (2k) + a22 (2k) + a12 (2k) + a21 (2k),

(22)

2a22(k) = a11 (2k) + a22 (2k) + a12 (2k + 2) + a21 (2k − 2),

(23)

2a12(k) = a11 (2k + 1) + a22 (2k + 3) + a12 (2k + 3) + a21 (2k + 1),

(24)

2a21(k) = a11 (2k − 1) + a22 (2k − 3) + a12 (2k − 1) + a21 (2k − 3).

(25)

Referring to Theorem 2, at least one, and therefore both prototiles Qi of the maximal compact solution have positive measure and cover Rd . In our notation that is a11 (0) 6= 0 and a22 (0) 6= 0. Employing (20) for k = 0, the first two equations become a11 (0) = a22 (0) + 2a12(0) and a22 (0) = a11 (0) + 2a12 (2). From this we conclude that a12(0) = a21 (0) = 0 and a11 (0) = a22(0).

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Since all entries are nonnegative, we see from (24) with k = 0 that a11 (1) = a22(3) = a12 (3) = a21 (1) = 0. In (22) k = 1 gives a11(2) = a22 (2) = a12 (2) = a21(2) = 0. Also, k = 1 in (25) yields a22 (1) = a12 (1) = 0. So far we have aij (k) = 0 for i, j = 1, 2 and k = ±1, ±2. To deal with |k| ≥ 3, we check the size of the prototiles. Set α = min T1 ∪ T2 and β = max T1 ∪ T2 . Then from (14) we conclude that α ≥ 0 and β ≤ 3. This implies that aij (k) = 0 for |k| ≥ 3 and by (19) Q is a self-affine Z-tiling set. In Example 10, where  A=

1 1

−1 1

 ,

AT1 = T1 ∪ T2 ,

and

AT2 = ((0, 1) + T1 ) ∪ ((1, 0) + T2 ),

(21) takes the form of the following four equations: 2a11(k) = a11 (Ak) + a22 (Ak) + a12(Ak) + a21 (Ak), 2a22(k) = a11 (Ak) + a22 (Ak) + a12(Ak + (1, −1)) + a21(Ak + (−1, 1)), 2a12(k) = a11 (Ak + (0, 1)) + a22 (Ak + (1, 0)) + a12(Ak + (1, 0)) + a21 (Ak + (0, 1)), 2a21(k) = a11 (Ak − (0, 1)) + a22 (Ak − (1, 0)) + a12(Ak − (1, 0)) + a21 (Ak − (0, 1)). ∈ Ti } be the (Euclidean) extension of Ti , then (17) implies √ Let δi = max{kxk : x √ −1/2 and 2δ1√= max(δ1 , δ2 ) and 2δ2 ≤ max(δ1 + 1, δ2 + 2), from which √ δ1 ≤ 1 + 2 δ2 ≤ 2 + 1. Therefore T1 ∪ T2 is contained in a disk of radius 2 + 1 and consequently aij (k) = 0 for |k| ≥ 5. In order to show that T1 ∪ T2 Z2 -tiles R2 , it is sufficient to show (19) for |k| ≤ 4. This is possible and an exercise in patience, but we shall take a more experimental approach and use the evidence from Fig. 4 that the maximum extension of both tiles in the x- and y-direction is 2. Using this geometric fact we have to verify (19) for k = (0, 0), (±1, 0), (0, ±1), ±(1, 1), ±(1, −1). As above, by recourse to Theorem 2 both prototiles Qi of the maximal compact solution have positive measure, that is a11(0, 0) 6= 0 and a22 (0, 0) 6= 0. Employing (20) for k = (0, 0), the first two equations become a11 (0, 0) = a22 (0, 0) + 2a12(0, 0) and a22 (0, 0) = a11(0, 0) + 2a12(0, 0). From this we conclude that a12 (0, 0) = a21 (0, 0) = 0 and |T1 | = a11 (0, 0) = a22 (0, 0) = |T2 |. Since all entries are nonnegative, we see from the equation for 2a12(0) that a11 (0, 1) = a22 (1, 0) = a12 (1, 0) = a21(0, 1) = 0 and by symmetry (20) a11(0, −1) = a22 (−1, 0) = a21 (−1, 0) = a12 (0, −1) = 0. Using a11 (0, 1) = a11 (1, 0) = 0 and (20), we obtain aij (1, −1) = 0 and aij (1, 1) = 0 for i, j = 1, 2. Next using a12(1, 0) = 0 implies a11(1, 0) = a21 (1, 0) = 0, whereas a21 (0, 1) = 0 gives a22 (0, 1) = a12 (0, 1) = 0. Together with the symmetry (20) this implies (19) for all k = (k1 , k2 ), |ki | ≤ 1 and that T1 ∪ T2 is a Z2 -tiling. 3. EXISTENCE OF SELF-AFFINE COLLECTIONS

Fix a dilation matrix A, a multiplicity M, and arbitrary finite digit sets 0ij ⊆ Zd , i, j = S 1, . . . , M. Set M i,j =1 0ij = 0 and write {1, . . . , M} = S. In this section we determine all solutions to the dilation equation (4). As in the one tile case, the solutions are described by digit expansions, the digits of which are taken from 0. Unlike the one tile case, not all

223

SELF-AFFINE TILINGS WITH SEVERAL TILES I

sequences of digits may occur as digit expansions. The difference is somewhat like that between the full sequence space on S and a subspace determined by a subshift of finite type [14]. To motivate the following definitions, observe that the dilation equations can be used to rewrite each prototile of a solution {Ti } in the form Ti = A

−1

M [

! 0ij + Tj .

j =1

Applying this operation again to each prototile Tj gives Ti =

M [

A

−1

0ij + A

−1

j =1

M [

! A

−1

0j k + A

−1

Tk

=

M M [ [

A−1 0ij + A−2 0j k + A−2 Tk .

j =1 k=1

k=1

Iterating this procedure n times shows that x ∈ Ti if and only if for some k ∈ S x∈

n X

A−j j + A−n Tk ,

(26)

j =1

where j ∈ 0ρj ρj+1 for j = 1, . . . , n and ρ1 = i. Since A−1 is contractive, all of the sets A−n Tk lie in a small ball for n large, and consequently x is essentially determined by the j . An elaboration of this argument yields P ROPOSITION 1. Set ( Qi =

x ∈R :x = d

∞ X

) A

−k

k , k ∈ 0ρk ρk+1 6= ∅ for some ρk ∈ S and ρ1 = i .

k=1

Q = {Qi }M i=1 is the unique solution to (4) for which all prototiles are nonempty and compact. As was observed in Examples 2 and 3, there are solutions for which some, but not all, of the sets are empty. Let T = {Ti } be any solution of (4) with Ti empty or compact. Define 6 ∅}. Removing the sets {Ti : i ∈ / N} in (4), we obtain another N = {i ∈ {1, . . . , M} : Ti = dilation equation for which the unique solution with all prototiles being nonempty and compact is defined as {Qi , i ∈ N} instead of i ∈ S. This remark leads to the following. /N D EFINITION. A nonempty subset N ⊆ S is said to be (A, 0)-closed, if j ∈ N and i ∈ imply 0ij = ∅. The definition is similar to the one from the theory of Markoff chains and will appear less coincidental in the next section. In this context closed sets are used to catalogue the nonempty prototiles in a solution to (4). For this we define the following sets of sequences in SN and 0 N . Given N ⊆ S, let  N RiN = (ρk )∞ k=1 ∈ S | ρ1 = i, ρk ∈ N, 0ρk ρk+1 6= ∅, ∀k

224 and let

GRÖCHENIG, HAAS, AND RAUGI

 ∞ N N N i = (k )k=1 ∈ 0 | k ∈ 0ρk ρk+1 for some ρ ∈ Ri

be the set of all “paths” in N starting at i. Then we define the sets ( ) ∞ X N −j N A j , (j ) ∈ i , Qi = x | x =

(27)

j =1 S / N . Set QN = {QN if i ∈ N and QN i = ∅ when i ∈ i }. Obviously S is closed, Qi = Qi , and QS = Q. It follows easily from the definition that if N is (A, 0)-closed then QN i 6= ∅ if and only if i ∈ N . Since the {Qi } may overlap, these are not always self-affine collections. See Example 5.

T HEOREM 1. (a) For any (A, 0)-closed set N , QN satisfies the dilation equation (4). The set QN i is compact if i ∈ N and empty otherwise. (b) If T = {Ti } is a collection of empty or compact sets that satisfies (4), then T = QN for some (A, 0)-closed set N . (c) If N1 and N2 are (A, 0)-closed sets and QN1 = QN2 , then N1 = N2 . Also, if N2 1 N1 ⊆ N2 then QN i ⊆ Qi for i = 1, . . . , M. In Example 2 the various (A, 0)-closed subsets of S are {1}, {2}, {1, 2}, {1, 3}, {1, 2, 3}, and S. They correspond to the solutions described in Section 2, in the respective orders. We assume from now on that an (A, 0)- closed set N ⊆ S is given. Choose a norm on Rd for which A−1 is a contraction; that is, kA−1 xk ≤ λkxk for P −k all x ∈ Rd and for some λ < 1. Define ω: 0 N → Rd by ω() = ∞ k=1 A k . With the N N product topology on 0 the map ω is continuous. Furthermore, for i ∈ N , N i ⊆ 0 is N N closed and hence compact, and therefore the set Qi = ω(i ) is also compact. We introduce further definitions and prove a lemma before proceeding with the proof of Theorem 1. The Euclidean metric on Rd is written d(·, ·). Let H∅ (Rd ) be the set containing the compact subsets of Rd and the empty set with the (modified) Hausdorff metric  max{supy∈Y d(X, y), supx∈X d(x, Y )} for X, Y 6= ∅ D(X, Y ) = for X = ∅. supy∈Y d(0, y) + 10 Let H∅M (Rd ) be the M-fold Cartesian product of H∅ (Rd ) with the product metric. Define the functions ϕi : H∅M → H∅ by ϕi (Z1 , . . . , ZM ) = A−1

M [

 0ij + Zj

(28)

j =1

and let ϕ: H∅M → H∅M be the product ϕ = (ϕ1 , . . . , ϕM ). ϕ n will denote the n-fold iterate n ) = Zn . of ϕ and we write ϕ n (Z1 , . . . , ZM ) = (Z1n , . . . , ZM 6 ∅} is L EMMA 1. Given any Z = (Z1 , . . . , ZM ) ∈ H∅M , suppose that N = {i | Zi = (A, 0)-closed. Then N (a) ϕ n (Z1 , . . . , ZM ) converges to (QN 1 , . . . , QM ) in the Hausdorff metric. M (b) If Y ∈ H∅ satisfies Zi = Yi whenever Yi 6= ∅, then for each n ∈ N and i = 1, . . . , M, ϕin (Z) ⊇ ϕin (Y ).

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SELF-AFFINE TILINGS WITH SEVERAL TILES I

Proof. (a) By (27) the sets QN i can be written in a concise form as follows: =

QN i

[

∞ X

ρ∈RiN

k=1

! A

−k

0ρk ρk+1 .

(29)

/ N then it is If i ∈ / N then RiN = ∅ and therefore QN i = ∅ as required. Furthermore, if i ∈ n clear from the definitions that Zi = ∅ for all n ∈ N and the convergence is assured in those components. We argue by induction that for all n ∈ N and i ∈ N, Zin

[

=

A

−n

Zρn+1 +

n X

! A

−k

0ρk ρk+1 .

(30)

k=1

ρ∈RiN

Before proceeding with the induction observe that the characterization of Zin given by (30) can be applied to prove convergence. Given α > 0 choose n0 > 0 so that for i = 1, . . . , M, and n > n0 , A−n Zi ⊆ B(0, α/2) and λn kω()k < α/2 for all  ∈ N i . Then for i ∈ N and n > n0 ! n ∞ [ X X
n N −n −k −k D Zi , Qi = D (A Zρn+1 + A 0ρk ρk+1 ), A 0ρk ρk+1 ρ∈RiN

≤ sup D A

k=1 −n

ρ∈RiN

!

∞ X

Zρn+1 ,

k=1

A

−k

0ρk ρk+1

< α.

(31)

k=n+1

N Therefore {Z n } converges to (QN 1 , . . . , QM ) as claimed. We now turn to the proof of (30). It is obvious for Zi0 . If it holds for each Zin , then by definition ! M [ 0ij + Zjn Zin+1 = A−1 j =1

=

M [

[

j =1

ρ∈RjN

A

−(n+1)

Zρn+1 +

n X

! A

−(k+1)

0ρk ρk+1 + A

−1

0ij .

(32)

k=1

Given ρ ∈ RjN with 0ij 6= ∅ define β = {βk }∞ k=1 by βk+1 = ρk for k ∈ N and β1 = i. Then by definition i ∈ N and β ∈ RiN . Conversely, if β ∈ RiN and ρ is defined as above, then ρ ∈ RjN with j = β2 . Thus (32) can be continued Zin+1

=

=

[

[

{j |0ij = 6 ∅}

ρ∈RjN

[

A

−(n+1)

β∈RiN

which completes the induction.

(A

−(n+1)

Zρn+1 +

n X

! A

−(k+1)

k=1

Zβn+2 +

n+1 X k=1

! A

−k

0βk βk+1

0ρk ρk+1 + A

−1

0ij

226

GRÖCHENIG, HAAS, AND RAUGI

(b) follows easily from the definition: ϕi (Y1 , . . . , YM ) = A

−1

M [

! 0ij + Yj

⊆A

−1

j =1

M [

! 0ij + Zj

j =1

= ϕi (Z1 , . . . , ZM ).  Proof of Theorem 1. We have already seen that the sets QN i are either compact or empty. Since ! ! ∞ ∞ [ X [ X N −k+1 −k+1 A 0ρk ρk+1 = A 0ρk+1 ρk+2 0ρ1 ρ2 + AQi = ρ∈RiN

=

[ 1≤j ≤M j ∈N

=

[

ρ∈RiN

k=1

0ij +

[

∞ X

ρ∈RjN

k=1

!!

k=1

A−k+1 0ρk ρk+1

M
[
0ij + QN = 0ij + QN j j

1≤j ≤M j ∈N

j =1

they satisfy the dilation equations. 6 ∅} and Let T be a collection of compact sets that satisfy (4). Define N = {i | Ti = suppose that i ∈ N and 0ni 6= ∅. Since Ti 6= ∅ Tn = A

−1

M [

! (0nj + Tj )

6= ∅

(33)

j =1

and therefore N is (A, 0)-closed. From the previous lemma ϕ n (T1 , . . . , TM ) converges to N (QN 1 , . . . , QM ) in the Hausdorff metric, but since (T1 , . . . , TM ) is a fixed point of ϕ we N have T = Q . If N1 and N2 are distinct (A, 0)-closed sets then without loss of generality there is N2 1 an n ∈ N1 \ N2 . It is immediate from the definition that QN n 6= ∅ while Qn = ∅. The N1 N2 inclusion Qi ⊆ Qi for N1 ⊆ N2 follows directly from the definition (27). Remark 1. Theorem 1 can be seen as giving a classification of the fixed points of the operator ϕ defined in (28). In contrast to the single tile case, in which ϕ is an iterated function system with a unique attracting fixed point [1], we may have a finite number of fixed points and periodic cycles, each having an associated basin of attraction. In Example 4 from Section 2 the set {(Q1 , ∅), (∅, Q2 )} is a period two cycle and any Z of the form (Z1 , ∅) or (∅, Z2 ) will accumulate at the cycle. The system also has the fixed point (Q1 , Q2 ) which is the limit of any Z with both components nonempty. Along with the trivial fixed point (∅, ∅) the above completely describes the limiting behavior of iterating ϕ in this example. We will be better equipped to describe this phenomenon later in the section. Following [16] we call {0ij , i, j = 1, . . . , M} = 0 a standard digit set if for each S j = 1, . . . , M, 0j = M i=1 0ij is a complete set of coset representatives for the group Zd /AZd . As we will see, it is precisely the standard digit sets that have relevance to wavelet theory.

SELF-AFFINE TILINGS WITH SEVERAL TILES I

227

T HEOREM 2. Suppose that 0 is a standard digit set and N is an (A, 0)-closed set. S N d Then QN is a self-affine collection and Zd + ( M i=1 Qi ) = R . Observe that these conditions are not necessary. In Example 6 0 is not a standard digit set, but Q is nevertheless a self-affine collection. It will follow from Theorem 3, that for such examples Q is not a Zd -tiling set. If N1 and N2 are two nonempty, disjoint (A, 0)-closed sets, then we infer from the S Ni d theorem that for i = 1, 2, the Zd -translates of the set M j =1 Qj cover R . Consequently we have: C OROLLARY 1. Suppose that 0 is a standard digit set and N1 and N2 are disjoint (A, 0)-closed sets. Then Q is not a Zd -tiling set. However, as is shown in Example 6, Q might tile with a coarser lattice. It is necessary to introduce some new ideas in the proof of Theorem 2. Let C be the M × M matrix, called the counting matrix, with entries cij = #0ij and let | det A| = q. The importance of C lies in the following lemma. L EMMA 2. (a) For any dilation A, digit set 0, and solution T to (4) q|Ti | ≤

M X

cij |Tj |

for i = 1, . . . , M.

(34)

j =1

Equality holds for all i, if and only if the disjointness property (5) is also satisfied and T is a self-affine collection. In particular for any self-affine collection T the column vector (|T1 |, . . . , |TM |)t is an eigenvector of C to the eigenvalue q. (b) If 0 is a standard digit set, then for each j = 1, . . . , M, M X

cij = q;

(35)

i=1

in other words, (1/q)C t is a stochastic matrix. Moreover, any solution T to (4) is automatically a self-affine collection. Proof. (a) For each i = 1, . . . , M, M M [ X 0ij + Tj ≤ cij |Tj |. q|Ti | = |ATi | = j =1

(36)

j =1

Equality holds in the ith equation if and only if the sets γ + Tj are essentially disjoint for S γ∈ M j =1 0ij . Thus, equality holds for all i if and only if T is a self-affine collection. SM P (b) The sum M i=1 cij represents the number of elements in the digit set 0j = i=1 0ij and thus it equals #0j = #Zd /AZd = | det A|. This gives M M X X i=1 j =1

cij |Tj | =

M X

q|Tj |.

(37)

j =1

This means that in (36) equality must hold for all i and thus T is a self-affine collection by (a).

228

GRÖCHENIG, HAAS, AND RAUGI

Proof of Theorem 2.

It remains to be proven that M [

Zd +

! = Rd .

QN i

(38)

i=1

We may choose compact sets Zi , i = 1, . . . , M, in Rd with the following properties: Zi = ∅ SM if and only if i ∈ / N , Zi ∩ Zj ∼ = ∅ for i 6= j , and i=1 Zi = [0, 1]d . n ), where ϕ was defined followRecall that we write ϕ n (Z1 , . . . , ZM ) = (Z1n , . . . , ZM ing (28). We will show by induction that the following statements are true for n ≥ 0: Zin = ∅

i∈ / N;

if and only if

(39)

Zin ∩ Zjn ∼ = ∅ for i 6= j ; M [

(40)

is a Zd -tile.

Zin

(41)

i=1

Since by Lemma 1 Zin converges to QN i in the Hausdorff metric, (38) follows from (41) as in [10]. The claim is argued by induction. It is certainly true for Z 0 = Z. Suppose (39)–(41) hold for Z n . Combining (40) and (41) we infer that for i 6= j or l 6= k, (k + Zin ) ∩ (l + Zjn ) ∼ = ∅. Since each 0j is a set of coset representatives, it follows that for i 6= j or l 6= k, γi ∈ 0i and γj ∈ 0j (γi + Ak + Zin ) ∩ (γj + Al + Zjn ) ∼ =∅ and M [ [

M  [ [

(γ + Zjn ) =

j =1 γ ∈0j

 (γ + Zjn )

γ ∈0ij

i,j =1

is an AZd -tile. Consequently, the sets A−1 (γ + Zjn ) are essentially disjoint for γ ∈ 0ij and 1 ≤ i, j ≤ M, and their union is a Zd -tile. Finally, since Zin+1

=A

−1

M  [ [ j =1

! (γ

+ Zjn )

γ ∈0ij

we conclude that Zin+1 ∩ Zjn+1 ∼ = ∅ for i 6= j and

SM

n+1 i=1 Zi

is a Zd -tile.

A different proof of Theorem 2 based on Fourier analytic methods will be given in Part II. 4. NECESSARY CONDITIONS AND MARKOFF CHAINS

We shall now discuss a number of necessary conditions for a dilation A and a digit set 0 to determine a self-affine tiling Q. Where not stated otherwise we take Q to be a self-affine Zd -tiling set. The setting will usually be further simplified by the assumption that for each i = 1, . . . , M, the set Qi has nonzero Lebesgue measure.

SELF-AFFINE TILINGS WITH SEVERAL TILES I

229

The importance of standard digit sets becomes apparent in the following theorem, which generalizes a similar result in the one-tile case [10]. T HEOREM 3. If Q is a Zd -tiling set and if Qi has positive Lebesgue measure for each i = 1, . . . , M, then 0 is a standard digit set. Proof. If 0 is not a standard digit set then, for some j , 0j is not a set of distinct coset representatives for Zd /AZd . Then either there are k1 ∈ 0i1 j and k2 = k1 − A` ∈ 0i2 j , S ` ∈ Zd , representing the same coset, or some coset is not represented in 0j = M i=1 0ij . In the former case by the self-similarity, k1 + Tj ⊆ ATi1 and k2 + Tj ⊆ ATi2 , which implies k1 + Tj ⊆ A(Ti2 + `). Since by assumption all Tj have positive measure, the inequality |Ti1 ∩ (` + Ti2 )| ≥ |A−1 (k1 + Tj )| > 0 furnishes a contradiction to the tiling property (1) and (3). Thus 0j consists of distinct P representatives of Zd /AZd . In particular, M i=1 cij ≤ q. SM Since the disjoint union T = i=1 Ti yields a Zd -tiling, T has measure 1 and Lemma 2 implies XX i=1 j =1

cij |Tj | = q

M X

|Ti | = q.

i=1

P P P P |Tj | = q provides a contradiction. This If M j =1 i=1 cij |Tj | < q i=1 cij < q, then S 0 is a complete set of representatives of Zd /AZd . means that M i=1 ij At this point, it is necessary to introduce concepts from the theory of Markoff chains which can be found in the standard texts, e.g., [5]. The results to which we make explicit reference appear as Proposition 2. Let P denote the stochastic matrix (1/q)C t . Then P is the matrix of transition probabilities for a Markoff chain with state space S. An invariant probability ν on S is a right eigenvector (ν(1), . . . , ν(M))T of P of eigenvalue 1 with PM i=1 ν(i) = 1. / N (see [5, remark on p. 384]). A set N ⊆ S is closed, if pj k = 0, whenever j ∈ N and k ∈ Since pj k 6= 0, if and only if 0kj 6= ∅, closed sets in the sense of Markoff chains coincide with the (A, 0)-closed sets defined in Section 3. An irreducible set is a closed set N ⊆ S that contains no proper closed subsets. Let R denote the union of all irreducible subsets of S and call a state x ∈ R recurrent (or persistent). The complement of R is the set I of transient states. Let pij(n) denote the (n)

ij th entry of the matrix P n . A state j is periodic of period τ if pjj = 0 unless n = mτ for m ∈ N, and τ is the smallest such integer. A Markoff chain is called aperiodic if S contains no periodic states and irreducible if S is irreducible. P ROPOSITION 2. (a) There exist disjoint irreducible sets R1 , . . . , Rk , with k > 0 so that S can be partitioned as S = R1 ∪ · · · ∪ Rk ∪ I.

(42)

(b) For each i, 1 ≤ i ≤ k, there is a unique invariant probability νi with the property / Ri , and νi (x) > 0 for x ∈ Ri . Every invariant probability is a linear that νi (x) = 0 for x ∈ combination of the νi ’s.

230

GRÖCHENIG, HAAS, AND RAUGI

(c) Suppose the Markoff chain is irreducible and some state x is periodic of period τ . Then every state is periodic of period τ and for the Markoff chain with the matrix of transition probabilities P τ , S is partitioned into τ nonempty irreducible subsets. (d) If a Markoff chain is irreducible and aperiodic then for all i, j ∈ S (n)

lim pij = ν({j }) > 0.

(43)

n→∞

where ν is the unique invariant probability on S = R1 . The proposition has the following consequence. T HEOREM 4. If 0 is a standard digit set and Q is a Zd -tiling set with |Qi | > 0 for 1 ≤ i ≤ M, then the associated Markoff chain is irreducible and aperiodic. Proof. By Corollary 1 only one irreducible set can appear in the decomposition (42). P −1 Set p = M i=1 |Qi |. It follows from Lemma 2 that the column vector ν with ν(i) = p |Qi | is a probability. As a result of the hypothesis that all prototiles have positive measure, ν(i) > 0 for all i ∈ S. In light of Proposition 2(b) the set of transient states I is empty and therefore in (42) S consists of a single recurrent set. Suppose that the chain is periodic. As in Example 4 it is possible to define the set of dilation equations A Qi = 2

M [ j =1

A0ij + AQj =

M [ j =1

A0ij +

M [ k=1

! 0kj + Qk

=

M [

2 0ik + Qk

(44)

k=1

2 . It is then possible to recursively define the dilation equations for some digit set 0ik SM n n A Qi = k=1 0ik + Qk , which we call eqn(n). For all n ∈ N the maximal solution QS of eqns(n) is then equal to the original solution Q of (4). Let C (n) denote the counting matrix of the dilation equations eqn(n). Using the fact that 0 is a standard digit set, a computation shows that C (n) = C n . Consequently, the equations determine a Markoff chain with transition matrix q −n (C (n) )t = P n , where P is the original transition matrix. By assumption there is a number τ ∈ N so that for the Markoff chain with transition matrix P τ , S contains τ nonempty irreducible subsets (Proposition 2(c)). Corollary 1 implies that QS = Q is not a Zd -tiling set.

A nonnegative matrix B is called primitive if, for some τ ∈ N, B τ has all positive entries. It follows from Proposition 2(d) that, under the hypothesis of the previous theorem, P must be primitive. Then certainly the original counting matrix C is also primitive. We have proved: C OROLLARY 2. If 0 is a standard digit set and Q is a Zd -tiling set with |Qi | > 0, 1 ≤ i ≤ M, then C is primitive. Using arguments from Markoff chains we can now further clarify the nature of the solutions of (4). The next theorem shows that for standard digit sets the tiles in QN are — up to null sets — either equal to the corresponding tiles in the maximal solution Q or equal to sets of measure zero. ∼ T HEOREM 5. If 0 is a standard digit set then for any (A, 0)-closed N ⊆ S, QN i = Qi , N / N. if i ∈ N , and Qi = ∅, if i ∈

SELF-AFFINE TILINGS WITH SEVERAL TILES I

231

Proof. If i ∈ N is recurrent, it belongs to a unique irreducible set L = Rj0 ⊆ N N appearing in the decomposition (42). Since, by Theorem 1 QL i ⊆ Qi ⊆ Qi , it will suffice L to show that |Qi | = |Qi |. According to Lemma 2, the vector with entries |QN i | is an eigenvector of P . Then | = 0 for all transient states i ∈ S. Proposition 2(b) implies that |QN i Let U ⊇ L be the smallest set with the property that if j ∈ U and k ∈ / U then 0j k = ∅. If j ∈ U \ L, then j is transient. This follows since, j ∈ U \ L if and only if there is a l ∈ L and a sequence j = ρ1 , . . . , ρm = l so that for each k = 1, . . . , m − 1, 0ρk ρk+1 6= ∅. Then, by the definition of irreducible, ρm−1 is a transient state. Furthermore, by the definition of recurrent, ρm−2 is transient and then, by induction, j is also transient. We conclude that for j ∈ U \ L, |Qj | = 0. Note that the equations of (4) express AQi for i ∈ U as a union of translates of Qj where the j also belong to U . Then the maximal solution Qˆ = {Qˆ i }i∈U of the equations ATi =

[

(0ij + Tj ) =

j ∈U

M [

(0ij + Tj )

j =1

ˆ i = Qi . satisfies Q In terms of the representation (26), for each i ∈ U , Qi = Qˆ i can be split into a disjoint union of sets, where the first contains all expansions with ρk ∈ L. Then Qi can be written as ∞ [ ∪ Ei,n , (45) Qi = QL i n=2

where ( Ei,n =

[

n−1 X

j ∈U \L

k=1

)

! A−k 0ρk ,ρk+1 +A−n Qj

Since |Qj | = 0, each Ei,n and therefore measure 0. It follows that |QL i | = |Qi |.

| ρ1 = i, ρn = j, ρk ∈ L, ∀1 ≤ k ≤ n−1 . S

n Ei,n

is a countable union of sets of

The next elementary theorem explains the problem that arises in cases like Example 7 with a = 1, b = 1, and c = 0. Note that no assumptions are made about the structure of the digit set. T HEOREM 6. Let A and 0 be arbitrary and assume that for each i ∈ S, Qi has positive measure. Suppose also that there is a nontrivial permutation σ of S for which the set of dilation equations ATσ (i) =

M [

0ij + Tσ (j )

(46)

j =1

with indices permuted is identical to the unpermuted set of dilation equations. Then the solution Q is not a lattice tiling set for any lattice 3. Proof. Choose i ∈ S for which σ (i) 6= i. Then since they are defined by the same digit expansions, Qi = Qσ (i) . By hypothesis Qi has positive Lebesgue measure and therefore

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|Qi ∩ Qσ (i) | > 0. Thus Q fails to satisfy the intersection property (2) and can never be a tiling set. It is possible to use the above information to give a complete analysis of the limiting behavior of points under iteration of the map ϕ defined in (28). Basically, looking at the decomposition of S in Proposition 2(a), aperiodic irreducible sets in the decomposition determine unique attracting fixed points, each with a well-defined basin of attraction. The irreducible sets that are not aperiodic fall under Proposition 2(c) and determine various attracting cycles, one corresponding to each of the the subgroups of the cyclic group of order τ . Each cycle has its own basin of attraction. We saw an example of this in the previous remark. Entries corresponding to transitive states will always converge to the empty set. The various behaviors combine to describe the general case. 5. WAVELET THEORY AND LATTICE TILINGS

There exists an interesting and at first glance surprising connection between wavelet theory and certain SATs of multiplicity 1 [10]. It is not surprising that similar results can be established for SATs with several tiles. This link between the geometric object of SATs and the analytic object of wavelet theory is important for several reasons: (a) It singles out — among all SATs which seem too complicated to be classified completely — a special class of SATs which are more accessible to a detailed analysis; (b) we will be able to apply methods from Fourier analysis and the theory of multiwavelets [7, 8, 13] to study SATs; (c) SATs will furnish a new class of examples of multiwavelet bases, of which only few concrete examples and constructions are known so far. We first recall that in wavelet theory one studies general approximation schemes, socalled MRAs. A multiresolution analysis V with respect to a dilation matrix A is a biinfinite sequence of closed subspaces Vj , j ∈ Z, of L2 (Rd ) with the following properties: • • • •

Vj ⊆ Vj +1 for all j ∈ Z. f (x) ∈ V0 if and only if f (x − k) ∈ V0 for all k ∈ Zd . f (x) ∈ V0 if and only if f (Aj x) ∈ Vj for j ∈ Z. V0 possesses an orthonormal basis of the form {φi (x − k), k ∈ Zd , i = 1, . . . , M}.

We refer to [4, 20] for background and construction procedures for MRAs. The number M of basis functions is called the multiplicity of the MRA. The φi ’s uniquely determine the MRA V and are said to generate the MRA. MRAs with multiplicity > 1 recently have become the object of intensive studies [7, 8, 13]. While most general results carry over from dimension 1 and multiplicity 1 to Rd and M > 1 without significant modifications, concrete examples are sparse. No generic example of an MRA with arbitrary multiplicity M is known for general dilation matrices. In this sense SATs contribute an interesting facet to wavelet theory. The following theorem is the counterpart of Theorem 1 in [10]. T HEOREM 7. (A) Suppose that T = {Ti , i = 1, . . . , M} is a self-affine Zd -tiling set with all prototiles of positive measure. Then the characteristic functions χTi , i = 1, . . . , M, generate a multiresolution analysis for L2 (Rd ).

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SELF-AFFINE TILINGS WITH SEVERAL TILES I

(B) Conversely, if a multiresolution analysis is generated by characteristic functions χTi , i = 1, . . . , M, then the T is a self-affine Zd tiling set. Moreover, the corresponding digits are a standard digit set. For the proof of the theorem we need a simple geometrical lemma first. L EMMA 3. Let T be a compact set in Rd whose Zd -translates are essentially disjoint. Q Then for any parallelepiped B = di=1 [ai , bi ] ⊆ Rd we have lim q −j

j →∞

X

|(k + T ) ∩ Aj B|2 = |T |2 |B|.

(47)

k∈Zd

Proof. Let Cρ (B) = {x ∈ Rd : kx − uk ≤ ρ for u ∈ ∂B} be the “collar” of thickness 2ρ around the boundary of B. Using kA−1 xk ≤ λkxk with λ < 1, we see that A−j Cρ (Aj B) ⊆ Cλj ρ (B) and thus for any ρ > 0 lim q −j |Cρ (Aj B)| = lim |A−j Cρ (Aj B)| = 0.

j →∞

j →∞

(48)

Equipped with this observation we partition the index set Zd in (47) into three subsets Aj = Zd ∩ (Aj B \ Cρ (Aj B)), Bj = Zd ∩ Cρ (Aj B), Cj = Zd \ (Aj ∪ Bj ). If we choose ρ = max{kxk : x ∈ T ∪ [0, 1]d }, then k ∈ Aj implies that k + T ⊆ Aj B and thus X |(k + T ) ∩ Aj B|2 = |T |2 q −j #Aj . q −j k∈Aj

P −j

j 2 2 −j Similarly, q k∈Bj |(k + T ) ∩ A B| ≤ |T | q #Bj . Finally, k ∈ Cj means |(k + T ) ∩ j A B| = 0 and the sum over Cj equals 0. To estimate #Aj , we observe that, by the choice of ρ,

Aj B \ C2ρ (Aj B) ⊆ Aj + [0, 1]d ⊆ Aj B. We combine the estimate
q −j |Aj B| − |C2ρ (Aj B)| ≤ q −j #Aj ≤ q −j |Aj B| = |B| with (48) and obtain limj →∞ q −j #Aj = |B|. Similarly, q −j #Bj ≤ q −j |C2ρ (Aj B)| → 0. We conclude that X |(k + T ) ∩ Aj B|2 = lim |T |2 q −j #Aj = |T |2 |B| lim q −j j →∞

as claimed.

k∈Zd

j →∞

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Proof of Theorem 7. (A) Suppose that T is a self-affine Zd -tiling set. Then we define ψij k (x) = q j/2 |Ti |−1/2 χTi (Aj x − k) and ( Vj =

f ∈ L (R ) | f (x) = d

2

)

M X X

(i) ak ψij k (x)

with a

(i)

∈ ` (Z ) . 2

d

(49)

i=1 k∈Zd

It is now easy to verify that V = (Vj )j ∈Z is an MRA of multiplicity M. The inclusions Vj ⊆ Vj +1 are an immediate consequence of the self-similarity (4), which amounts to the scaling relations χTi (x) =

M X X

χTj (Ax − k).

(50)

j =1 k∈0ij

The existence of the orthonormal basis {|Ti |−1/2 χTi (x − k), k ∈ Zd , i = 1, . . . , M} for V0 and the translation invariance of V0 are guaranteed by the definition. S We only have to show that j ∈Z Vj is dense in L2 (Rd ). For this we first compute P P 2 d the orthogonal projection Pj f = M k∈Zd hf, ψij k iψij k from L (R ) onto Vj for the i=1 characteristic function f = χB of a parallelepiped B. In this case hχB , ψij k i = q −j/2 |Ti |−1/2 |(k + Ti ) ∩ Aj B| and thus kPj χB k22 =

M X

PM

i=1 |Ti | = 1.

|(k + Ti ) ∩ Aj B|2 .

(51)

k∈Zd

i=1

Since T is a Zd -tiling,

X

q −j |Ti |−1

Therefore, by Lemma 3,

lim kPj χB k22 =

j →∞

M X

|Ti ||B| = kχB k22 .

i=1

But since characteristic functions of parallelepipeds span a dense subspace of L2 (Rd ) and S since kf − Pj f k22 = kf k22 − kPj f k22 , this suffices to show that Vj is dense in L2 (Rd ). We have verified that (Vj )j ∈Z is a multiresolution analysis. (B) Now assume that the functions χTi generate a MRA. Then the orthogonality of the basis functions gives immediately Z |Ti |δij δkl =

Rd

χTi (x − k)χTj (x − l) dx = |(k + Ti ) ∩ (l + Tj )|,

in other words, the disjointness of the tiles. Next, since χTi (A−1 x) ∈ V−1 ⊆ V0 , it can be expressed in terms of the orthonormal basis of V0 in the form of a so-called scaling relation: χATi (x) = χTi (A−1 x) =

M X X j =1 k∈Zd

cj k χTj (x − k)

for i = 1, . . . , M,

(52)

SELF-AFFINE TILINGS WITH SEVERAL TILES I

235

R where the coefficients cj k = χATi (x)χk+Tj (x) dx can only take the values 0 or 1, because the integer translates of the prototiles are disjoint. If for each i we denote the set of translates k ∈ Zd for which cj k = 1 by 0ij , then (52) can be rewritten as the following self-similarity of sets ATi ∼ =

M [

(0ij + Tj )

for i = 1, . . . , M.

j =1

Next we deduce that the Ti tile with lattice Zd . S Since Vj is dense in L2 (Rd ), we know that kPj χB k22 → kχB k22 = |B| for any parallelepiped B. On the other hand, from (51) we know that lim kPj χB k22 = |B|

j →∞

M X

|Ti |.

i=1

P It follows that M i=1 |Ti | = 1. Now consider the function 8(x) =

M X X

χTi (x − k).

i=1 k∈Zd

Since the prototiles are pairwise disjoint, 0 ≤ 8(x) ≤ 1. Then Z

Z [0,1]d

8(x) dx =

M X Rd

i=1

! χTi (x) dx =

M X

|Ti | = 1.

i=1

We see that 8(x) = 1, which is equivalent to the Zd -tiling property of T . It follows now from Theorem 3 that 0 is a standard digit set, and the theorem is proved completely. As a consequence we can construct orthonormal wavelet bases with compact support, but without smoothness, starting from SATs. T HEOREM 8. Suppose that T is a self-affine Zd -tiling set. Then there exist (q − 1)M S functions ψl with compact support in M j =1 Tj , such that  j/2 q ψl (Aj x − k), j ∈ Z, k ∈ Zd , l = 1, . . . , (q − 1)M

(53)

is an orthonormal basis for L2 (Rd ). The ψl can be written explicitly as linear S combinations of the functions χTi (Ax − k), k ∈ 0ij . Proof. Following the standard line of arguments we have to find an orthonormal basis of the form {ψl (x − k), k ∈ Zd , l = 1, . . . , (q − 1)M} in W0 := V1 V0 , the orthogonal L S complement of V0 in V1 . Since j ∈Z Wj = j ∈Z Vj = L2 (Rd ), the collection (53) is then an orthonormal basis for L2 (Rd ). We refer to [4, 20] for the general construction of wavelet bases.

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By definition (49) f ∈ V1 if f (x) =

M X X

aik q 1/2|Ti |−1/2 χTi (Ax − k)

i=1 k∈Zd

with (aik )k∈Zd ∈ `2 (Zd ), and f ∈ W0 , if and only if f ⊥ χTj (x − l). Rewriting (52) as χTj (x − l) =

   M X  X q −1/2 q 1/2 χTr (A(x − l) − m), |Tr | |Tr | r=1 m∈0jr

we calculate for f ∈ W0 0 = hf, χTj (. − l)i =

 M X X  M X X |Tr | 1/2 δir δk,Al+m aik q d i=1 r=1 m∈0jr k∈Z

 M X  X |Tr | 1/2 = ar,Al+m . q

(54)

r=1 m∈0jr

Now consider the M linear equations M X X

|Tr |1/2arm = 0

(55)

r=1 m∈0jr

PM P in the M j =1 r=1 #0j r = qM variables arm . It is not hard to check that these equations are linearly independent. Consequently the null space has dimension qM − M. Choose an orthonormal basis (u(s) rm ), s = 1, . . . , qM − M, for the null space and define the functions   M X M X X q 1/2 (s) uim χTi (Ax − m). (56) ψs (x) = |Ti | Then supp ψs ⊆

i=1 j =1 m∈0ji

SM

j =1 Tj .

From these support properties the orthogonality relations

hψs , χTj (. − l)i = 0 = hψs , ψs 0 (. − l)i for l 6= 0 or s 6= s 0 are clear. Since by (54) hψs , χTj i =

M X X

1/2 u(s) = 0, rm (|Tr |/q)

r=1 m∈0jr

we see that ψs ∈ W0 . Since hψs , ψs 0 i =

M X X

0

(s ) u(s) rm urm = δss 0 ,

r=1 m∈0jr

the functions {ψs (x −k), k ∈ Zd , s = 1, . . . , qM −M} form an orthonormal system in W0 .

SELF-AFFINE TILINGS WITH SEVERAL TILES I

237

We finally show that this orthonormal system is complete in W0 . Suppose that f ∈ W0 is orthogonal to this basis. Then the coefficients of f satisfy (54) and for all l and s hf, ψs (. − l)i =

M X X

ak,Al+m u(s) rm = 0.

r=1 m∈0jr

The vectors u(s) are by definition an ONB for the null space in (54) and thus for all l ∈ Zd we have ak,Al+m = 0. In other words, f = 0 and the orthonormal system is complete in W0 . A more explicit solution can be obtained as follows: First find an orthonormal basis of P P vectors u = (urm ), satisfying M m∈0jr urm = 0 for a fixed j and define for each such r=1 u the function ψu (x) =

M X X

urm (q/|Tr |)1/2χTr (Ax − m).

r=1 m∈0jr

P Then supp ψu ⊆ Tj . Counting dimensions, for each j there are exactly M r=1 cj r − 1 such PM PM functions. Doing this for each j , we get a collection of j =1 r=1 crm − M = qM − M functions ψu . As above they form an orthonormal basis for W0 . In the context of multiwavelet theory it is therefore of interest to know when the construction of an SAT, starting from a standard digit set yields a Zd -tiling of Rd . The examples indicate that two phenomena may contribute to a failure. (a) The prototiles Ti tile with a coarser lattice than Zd . See Example 7 with a = b = c = 2. If the multiplicity is one, this is the only obstruction [3, 18]. (b) The redundant case: Rd can already be tiled by Zd with a smaller number N < M of tiles. See Examples 3 and 7 with a = b = 1, c = 0. This is a genuinely new phenomenon in the case of higher multiplicity. We conjecture that a combination of these two cases is all that can go wrong. The fundamental question in this context is to decide which choices of standard digit sets generate Zd -tilings. This is a difficult and subtle question even in the case of only one tile; see [2, 3, 9, 16, 17, 18]. We shall return to this question in Part II. ACKNOWLEDGEMENT This research began while the third author visited the Department of Mathematics at the University of Connecticut. Its support and hospitality are gratefully acknowledged. The second author thanks the University of Washington for providing access to their research facilities while this paper was in preparation.

REFERENCES 1. M. Barnsley, “Fractals Everywhere,” Academic Press, New York, 1988. 2. D. Cerveau, J.-P. Conze, and A. Raugi, Fonctions harmoniques pour un opérateur de transition en dimension > 1, Bol. Soc. Bras. Mat. 27 (1996), 1–26. 3. J.-P. Conze, L. Hervé, and A. Raugi, Pavages auto-similaires, opérateurs de transfert et critères de réseau, Bol. Soc. Bras. Mat. 28 (1997), 1–42.

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4. I. Daubechies, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, Philadelphia, 1992. 5. W. Feller, “An Introduction to Probability Theory and its Applications,” Vol. I, Wiley, New York, 1968. 6. T. Flaherty and Y. Wang, Haar-type multiwavelet bases and self-affine multi-tiles, preprint, March 1997. 7. J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), 373–401. 8. T. Goodman and S. L. Lee, Wavelets of multiplicity r, Trans. Amer. Math. Soc. 342 (1994), 307–324. 9. K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131–170. 10. K. Gröchenig and W. Madych, Multiresolution analysis, Haar bases, and self-similar tilings, IEEE Trans. Inform. Theory 38, Part 2 (1992), 556–568. 11. B. Grünbaum and G. C. Shephard, “Tilings and Patterns,” Freeman, New York, 1987. 12. D. Hacon, N. C. Saldanha, and J. P. Veerman, Remarks on self-affine tilings, Experiment. Math. 3 (1994), 317–326. 13. Q. Jiang, Multivariate matrix refinable functions with matrix dilations, preprint, 1997. 14. M. Keane, Ergodic theory and subshifts of finite type, in “Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces” (T. Bedford, M. Keane, and C. Series, Eds.), Oxford Univ. Press, Oxford, 1991. 15. R. Kenyon, Self-replicating tilings, in “Symbolic Dynamics and Applications” (P. Walters, Ed.), Contemp. Math. 135 (1992), 239–264. 16. J. Lagarias and Y. Wang, Integral self-affine tilings in Rn , I, J. London Math. Soc. 53 (1996), 161–179. 17. J. Lagarias and Y. Wang, Haar type wavelet bases in Rn and algebraic number theory, J. Number Theory 57 (1996), 181–197. 18. J. C. Lagarias and Y. Wang, Integral self-affine tiles in Rn . II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83–102. 19. B. Mandelbrot, “The Fractal Geometry of Nature,” Freeman, New York, 1983. 20. Y. Meyer, “Ondelettes et Opérateurs I, II,” Hermann, Paris, 1990. 21. R. Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10 (1974), 266–271. 22. C. Radin, Space tilings and substitutions, Geom. Dedicata 55 (1995), 257–264. 23. R. S. Strichartz, Wavelets and self-affine tilings, Constructive Approx. 9 (1993), 283–297. 24. W. P. Thurston, Groups, tilings, and finite state automata, in “Lecture Notes,” Summer Meeting of the Amer. Math. Soc., Boulder, 1989. 25. A. Vince, Replicating tesselations, SIAM J. Discrete Math. 6 (1993), 501–521.